Assume that \(\big (X_{\epsilon }, \mathsf {d}_{\epsilon }, \mathcal {H}^5\big )\)  is an \(\operatorname{RCD}(K_1, \infty )\)  space for some \(K_1 \in \mathbb {R}\) . Note that it is easy to see that an open set \(B_R(p) \setminus \overline{B}_r(p)\)  for all \(r, R \in (0, \infty )\)  with \(r<R\)  is isometric as Riemannian manifolds to an open subset of a closed Riemannian manifold (by gluing “two caps” along the boundary). In particular the localities of the minimal relaxed slope, of the Laplacian and of the Hessian (see [1]}) yield
\(\int _{X_{\epsilon }}\langle \nabla f, \nabla \phi \rangle \mathop {}\!\mathrm {d}\mathcal {H}^5=-\int _{X_{\epsilon }}\operatorname{tr}(\mathrm {Hess}_f) \phi \mathop {}\!\mathrm {d}\mathcal {H}^5, \qquad \forall \, f \in D(\Delta ),\quad \forall \, \phi \in C^{\infty }_c(X_{\epsilon } \setminus \lbrace p\rbrace ).\) 
